Theoretical Fluid Dynamics

by Shivamoggi, Bhimsen K.

• ISBN13: 9780471056591
• ISBN10: 0471056596
• Edition: 2nd
• Format: Hardcover
• Copyright: 1998-01-29
• Publisher: Wiley-Interscience

Summary

"Although there are many texts and monographs on fluid dynamics, I do not know of any which is as comprehensive as the present book. It surveys nearly the entire field of classical fluid dynamics in an advanced, compact, and clear manner, and discusses the various conceptual and analytical models of fluid flow." - Foundations of Physics on the first edition Theoretical Fluid Dynamics functions equally well as a graduate-level text and a professional reference. Steering a middle course between the empiricism of engineering and the abstractions of pure mathematics, the author focuses on those ideas and formulations that will be of greatest interest to students and researchers in applied mathematics and theoretical physics. Dr. Shivamoggi covers the main branches of fluid dynamics, with particular emphasis on flows of incompressible fluids. Readers well versed in the physical and mathematical prerequisites will find enlightening discussions of many lesser-known areas of study in fluid dynamics. This thoroughly revised, updated, and expanded Second Edition features coverage of recent developments in stability and turbulence, additional chapter-end exercises, relevant experimental information, and an abundance of new material on a wide range of topics, including: * Hamiltonian formulation * Nonlinear water waves and sound waves * Stability of a fluid layer heated from below * Equilibrium statistical mechanics of turbulence * Two-dimensional turbulence

Author Biography

BHIMSEN K. SHIVAMOGGI, PhD, is a professor in the Department of Mathematics at the University of Central Florida.

Table of Contents

Preface	xvii	(2)
Acknowledgments	xix
Chapter 1. Review of Basic Concepts and Equations of Fluid Dynamics	1	(26)
1.1 Introduction to Fluid Dynamics	1	(21)
Fluid Model of Systems	1	(1)
The Objective of Fluid Dynamics	1	(1)
The Fluid State	2	(1)
Description of the Flow Field	3	(1)
Volume Forces and Surface Forces	4	(3)
Relative Motion Near a Point	7	(3)
Stress-Strain Relations	10	(1)
Equations of Fluid Flows	11	(7)
The Transport Theorem	12	(1)
The Material Derivative	13	(1)
The Law of Conservation of Mass	13	(1)
Equation of Motion	13	(1)
The Energy Equation	14	(2)
The Equation of Vorticity	16	(1)
The Incompressible Fluid	16	(2)
Hamiltonian Formulation of Fluid-Flow Problems	18	(4)
Hamiltonian Dynamics of Continuous Systems	18	(3)
Three-Dimensional Incompressible Flows	21	(1)
1.2 Surface Tension	22	(3)
Capillary Rises in Liquids	24	(1)
1.3 A Program for Analysis of the Governing Equations	25	(2)
Chapter 2. Dynamics of Inviscid Incompressible Fluid Flows	27	(186)
2.1 Fluid Kinematics and Dynamics	27	(15)
Stream Function	27	(2)
Equations of Motion	29	(1)
Integrals of Motion	29	(1)
Capillary Waves on a Spherical Drop	30	(2)
Cavitation	32	(1)
Rates of Change of Material Integrals	33	(2)
Irrotational Flow	35	(1)
Simple-Flow Patterns	36	(6)
The Source Flow	36	(2)
The Doublet Flow	38	(1)
The Vortex Flow	39	(1)
Doublet in a Uniform Stream	39	(1)
Uniform Flow Past a Circular Cylinder with Circulation	40	(2)
2.2 The Complex-Variable Method	42	(22)
The Complex Potential	42	(3)
Conformal Mapping of Flows	45	(6)
Hydrodynamic Images	51	(2)
Principles of Free-Streamline Flow	53	(11)
Schwartz-Christoffel Transformation	53	(6)
Hodograph Method	59	(5)
2.3 Three-Dimensional Irrotational Flows	64	(12)
Special Singular Solutions	64	(4)
The Source Flow	65	(1)
The Doublet Flow	66	(2)
d'Alembert's Paradox	68	(1)
Image of a Source in a Sphere	69	(2)
Flow Past an Arbitrary Body	71	(2)
Unsteady Flows	73	(1)
Added Mass of Bodies Moving Through a Fluid	74	(2)
2.4 Vortex Flows	76	(21)
Vortex Tubes	76	(1)
Induced Velocity Field	77	(1)
Biot-Savart's Law	78	(5)
Vortex Ring	83	(4)
Hill's Spherical Vortex	87	(2)
Vortex Sheet	89	(2)
The Vortex Breakdown: Brooke Benjamin's Theory	91	(6)
2.5 Rotating Flows	97	(17)
Governing Equations and Elementary Results	97	(1)
Taylor-Proudman Theorem	98	(1)
Propagation of Waves in a Rotating Fluid	99	(2)
Plane Inertial Waves	101	(2)
Forced Wavemotion in a Rotating Fluid	103	(3)
The Elliptic Case	104	(1)
The Hyperbolic Case	105	(1)
Slow Motion Along the Axis of Rotation	106	(4)
Rossby Waves	110	(4)
2.6 Water Waves	114	(54)
Governing Equations	114	(2)
Surface Waves in a Semi-infinite Liquid	116	(1)
Surface Waves in a Liquid Layer of Finite Depth	116	(2)
Shallow-Water Waves	118	(2)
Water Waves Generated by an Initial Displacement Over a Localized Region	120	(3)
Water Waves Generated by a Finite Train of Harmonic Waves	123	(2)
Waves on a Steady Stream	125	(5)
One-Dimensional Gravity Waves	126	(1)
One-Dimensional Capillary-Gravity Waves	127	(1)
Ship Waves	127	(3)
Gravity Waves in a Rotating Fluid	130	(3)
Theory of Tides	133	(2)
Nonlinear Shallow Water Waves	135	(23)
Solitary Waves	137	(3)
Periodic Cnoidal Waves	140	(5)
Interacting Solitary Waves	145	(3)
Stokes Waves	148	(2)
Modulational Instability and Envelope Solitons	150	(8)
Nonlinear Resonant Three-Wave Interactions of Capillary-Gravity Waves	158	(5)
Second-Harmonic Resonance	163	(3)
Hydraulic Jump	166	(2)
2.7 Applications to Aerodynamics	168	(45)
Airfoil Theory: Method of Complex Variables	169	(13)
Force and Moments on an Arbitrary Body	169	(2)
Flow Past an Arbitrary Cylinder	171	(3)
Flow Around a Flat Plate	174	(1)
Flow Past an Airfoil	175	(3)
The Joukowski Transformation	178	(4)
Thin Airfoil Theory	182	(13)
Thickness Problem	184	(2)
Camber Problem	186	(4)
Flat Plate at an Angle of Attack	190	(1)
Combined Aerodynamic Characteristics	191	(1)
The Leading-Edge Problem of a Thin Airfoil	192	(3)
Slender-Body Theory	195	(1)
Lifting-Line Theory for Wings	196	(5)
Oscillating Thin-Airfoil Theory: Theodorsen's Theory	201	(12)
Chapter 3. Dynamics of Inviscid Compressible Fluid Flows	213	(102)
3.1 Review of Thermodynamics	213	(7)
Thermodynamic System and Variables of State	213	(1)
The First Law of Thermodynamics; Reversible and Irreversible Processes	214	(2)
The Second Law of Thermodynamics	216	(2)
Liquid and Gaseous Phases	218	(1)
Application of Thermodynamics to Fluid Flows	219	(1)
3.2 Isentropic Flows	220	(4)
The Energy Equation	220	(1)
Stream-Tube Area and Velocity Relations	221	(3)
3.3 Shock Waves	224	(15)
The Normal Shock Wave	224	(8)
The Oblique Shock Wave	232	(2)
Blast Waves: Sedov's Solution	234	(5)
3.4 Flows with Heat Transfer	239	(4)
Rayleigh Flow	239	(1)
Detonation and Deflagration Waves	240	(3)
3.5 Potential Flows	243	(20)
Governing Equations	243	(1)
Streamline Coordinates	244	(2)
Conical Flows	246	(2)
Small Perturbation Theory	248	(1)
Characteristics	249	(14)
A Singular-Perturbation Problem for Hyperbolic Systems	254	(9)
3.6 The Hodograph Method	263	(10)
The Hodograph Transformation	263	(4)
The Lost Solution	267	(1)
The Limit Line	268	(5)
3.7 Nonlinear Theory of Plane Sound Waves	273	(19)
Riemann Invariants	274	(6)
Nonlinear Propagation of a Sound Wave	280	(2)
Nonlinear Resonant Three-Wave Interactions of Sound Waves	282	(5)
Burgers' Equation	287	(5)
3.8 Applications to Aerodynamics	292	(23)
Thin Airfoil Theory	292	(23)
Thin Airfoil in a Linearized Supersonic Flow	292	(2)
Far-Field Behavior of Supersonic Flow Past a Thin Airfoil	294	(3)
Thin Airfoil in Transonic Flows	297	(2)
Slender Bodies of Revolution	299	(7)
Oscillating Thin Airfoil in Subsonic Flows: Possio's Theory	306	(6)
Oscillating Thin Airfoils in Supersonic Flows: Stewartson's Theory	312	(3)
Chapter 4. Dynamics of Viscous Fluid Flows	315	(78)
4.1 Exact Solutions to Equations of Viscous Fluid Flow	315	(20)
Channel Flows	315	(2)
Decay of a Line Vortex	317	(2)
Line Vortex in a Uniform Stream	319	(1)
Diffusion of a Localized Vorticity Distribution	320	(4)
Flow Due to a Suddenly Accelerated Plane	324	(1)
The Round Laminar Jet: Landau's Solution	325	(3)
Ekman Layer at a Free Surface in a Rotating Fluid	328	(2)
Centrifugal Flow Due to a Rotating Disk	330	(2)
Shock Structure: Becker's Solution	332	(2)
Couette Flow of a Gas	334	(1)
4.2 Flows at Low Reynolds Numbers	335	(15)
Dimensional Analysis	336	(1)
Stokes' Flow Past a Rigid Sphere	336	(4)
Stokes' Flow Past a Spherical Drop	340	(2)
Stokes' Flow Past a Rigid Circular Cylinder	342	(1)
Oseen's Flow Past a Rigid Sphere	343	(3)
Oseen's Approximation for Periodically Oscillating Wakes	346	(4)
4.3 Flows at High Reynolds Numbers	350	(31)
Prandtl's Boundary-Layer Concept	350	(1)
The Method of Matched Asymptotic Expansions	351	(4)
Location and Nature of Boundary Layers	355	(3)
Incompressible Flow Past a Flat Plate	358	(7)
The Outer Expansion	359	(1)
The Inner Expansion	360	(4)
Flow Due to Displacement Thickness	364	(1)
Separation of Flow in a Boundary Layer: Landau's Theory	365	(3)
Boundary Layers in Compressible Flows	368	(4)
Crocco's Integral	370	(1)
Flow Past a Flat Plate: Howarth-Dorodnitsyn Transformation	371	(1)
Flow in a Mixing Layer Between Two Parallel Streams	372	(4)
Geometrical Characteristics of the Mixing Flow	374	(2)
Narrow Jets	376	(1)
Wakes	377	(1)
Periodic Boundary-Layer Flows	378	(3)
4.4 Jeffrey-Hamel Flow	381	(12)
The Exact Solution	382	(4)
Flows at Low Reynolds Numbers	386	(4)
Flows at High Reynolds Numbers	390	(3)
Chapter 5. Hydrodynamic Stability	393	(94)
5.1 Introduction to Hydrodynamic Stability	393	(1)
5.2 Thermal Instability of a Layer of Fluid Heated from Below	394	(12)
The Characteristic-Value Problem	394	(5)
The Variational Problem	399	(4)
Nonlinear Effects	403	(3)
Instability of the Roll Pattern	406	(1)
5.3 Stability of Couette Flow	406	(14)
Inviscid Couette Flow: Rayleigh Criterion	406	(4)
Heuristic Derivation	406	(2)
Rigorous Derivation	408	(2)
Couette Flow with Axial Velocity: Howard-Gupta Theory	410	(2)
Viscous Couette Flow: Synge's Theory	412	(7)
Nonlinear Effects	419	(1)
Instability of the Toroidal Vortices	419	(1)
5.4 Rayleigh-Taylor Instability of Superposed Fluids	420	(6)
The Linear Problem	420	(3)
The Nonlinear Problem	423	(3)
5.5 Kelvin-Helmholtz Instability	426	(9)
The Stratified Fluid in Nonuniform Streaming	427	(1)
The Case of Two Uniform Fluids in Relative Motion Parallel to the Plane Interface	428	(1)
A Shear Layer in a Stratified Fluid	429	(3)
Stability of an Interface Between a Liquid and a Gas Stream	432	(3)
Subsonic Gas Flow	434	(1)
Supersonic Gas Flow	434	(1)
5.6 Capillary Instability of a Liquid Jet	435	(4)
5.7 Stability of Parallel Flows	439	(48)
The Orr-Sommerfeld Equation	440	(2)
The Inviscid Solutions	442	(3)
The Initial-Value Problem: Case-Dikii Theory	445	(6)
Inviscid Stability Theory	451	(14)
Discontinuities in the Mean Flow	451	(2)
Odd and Even Solutions	453	(1)
Self-Excited and Damped Disturbances	454	(1)
Local and Global Necessary Conditions for the Existence of Non-neutral and Neutral Modes	455	(3)
Howard's Semicircle Theorem	458	(2)
Sufficient Conditions for the Existence of Self-Excited and Neutral Modes	460	(5)
Viscous Theory	465	(10)
Heisenberg Criterion	468	(2)
General Characteristics of the Neutral-Stability Curve	470	(3)
Computation of the Neutral-Stability Curve	473	(2)
Nonlinear Theory	475	(1)
Arnol'd Stability Approach	476	(11)
Hamiltonian Formulation of Two-Dimensional Incompressible Flows	479	(3)
Arnol'd Stability of Two-Dimensional Incompressible Flows	482	(5)
Chapter 6. Dynamics of Turbulence	487	(48)
6.1 The Origin and Nature of Turbulence	487	(2)
6.2 Three-Dimensional Turbulence	489	(31)
A Statistical Formalism	489	(1)
The Probability Density	490	(3)
The Autocorrelation	493	(3)
The Central Limit Theorem	496	(3)
Symmetry Conditions	499	(1)
Spectral Theory	500	(3)
Heisenberg's Theory	503	(3)
Kolmogorov's Universal Equilibrium Theory	506	(1)
Equilibrium Statistical Mechanics: Lee's Theory	507	(3)
Homogeneous, Isotropic Turbulence: Taylor's Correlation Theory	510	(8)
The von Karman-Howarth Equation	518	(2)
6.3 Two-Dimensional Turbulence	520	(11)
Conserved Quantities for a Two-Dimensional Flow	521	(1)
Fourier Analysis of the Turbulent Velocity Field	522	(1)
Energy and Enstrophy Cascades	523	(2)
Self-Organization and Self-Degradation in Two-Dimensional Flows	525	(2)
Batchelor-Kraichnan Theory of the Inertial Ranges	527	(1)
Equilibrium Statistical Mechanics: Kraichnan's Theory	528	(3)
6.4 Turbulent Dispersion: Lin's Theory	531	(4)
Bibliography	535	(12)
Subject Index	547